/*
Copyright 2008-2026
Matthias Ehmann,
Carsten Miller,
Andreas Walter,
Alfred Wassermann
This file is part of JSXGraph.
JSXGraph is free software dual licensed under the GNU LGPL or MIT License.
You can redistribute it and/or modify it under the terms of the
* GNU Lesser General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version
OR
* MIT License: https://github.com/jsxgraph/jsxgraph/blob/master/LICENSE.MIT
JSXGraph is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public License and
the MIT License along with JSXGraph. If not, see <https://www.gnu.org/licenses/>
and <https://opensource.org/licenses/MIT/>.
*/
/*global JXG: true, define: true*/
/*jslint nomen: true, plusplus: true*/
/*eslint no-loss-of-precision: off */
import Mat from "./math.js";
import Type from "../utils/type.js";
/**
* The JXG.Math.ProbFuncs namespace bundles various probability functions, e.g. error function,
* see {@link https://en.wikipedia.org/wiki/Error_function}.
* Ported from
* by {@link https://github.com/jeremybarnes/cephes/blob/master/cprob/ndtr.c}
*
* ```
* Cephes Math Library Release 2.9: November, 2000
* Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
* ```
*
* @name JXG.Math.ProbFuncs
* @exports Mat.ProbFuncs as JXG.Math.ProbFuncs
* @namespace
*/
Mat.ProbFuncs = {
MAXNUM: 1.701411834604692317316873e38, // 2**127
SQRTH: 7.07106781186547524401e-1, // sqrt(2)/2
SQRT2: 1.4142135623730950488, // sqrt(2)
MAXLOG: 7.08396418532264106224e2, // log 2**1022
P: [
2.46196981473530512524e-10, 5.64189564831068821977e-1, 7.46321056442269912687,
4.86371970985681366614e1, 1.96520832956077098242e2, 5.26445194995477358631e2,
9.3452852717195760754e2, 1.02755188689515710272e3, 5.57535335369399327526e2
],
Q: [
1.32281951154744992508e1, 8.67072140885989742329e1, 3.54937778887819891062e2,
9.75708501743205489753e2, 1.82390916687909736289e3, 2.24633760818710981792e3,
1.65666309194161350182e3, 5.57535340817727675546e2
],
R: [
5.64189583547755073984e-1, 1.27536670759978104416, 5.01905042251180477414,
6.16021097993053585195, 7.4097426995044893916, 2.9788666537210024067
],
S: [
2.2605286322011727659, 9.39603524938001434673, 1.20489539808096656605e1,
1.70814450747565897222e1, 9.60896809063285878198, 3.3690764510008151605
],
T: [
9.60497373987051638749, 9.00260197203842689217e1, 2.23200534594684319226e3,
7.00332514112805075473e3, 5.55923013010394962768e4
],
U: [
3.35617141647503099647e1, 5.21357949780152679795e2, 4.59432382970980127987e3,
2.26290000613890934246e4, 4.92673942608635921086e4
],
// UTHRESH: 37.519379347,
M: 128.0,
MINV: 0.0078125,
/**
*
* Exponential of squared argument
*
* ```
* SYNOPSIS:
*
* double x, y, expx2();
* int sign;
*
* y = expx2( x, sign );
*
* DESCRIPTION:
*
* Computes y = exp(x*x) while suppressing error amplification
* that would ordinarily arise from the inexactness of the
* exponential argument x*x.
*
* If sign < 0, the result is inverted; i.e., y = exp(-x*x) .
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -26.6, 26.6 10^7 3.9e-16 8.9e-17
* ```
*
* @private
* @param {Number} x
* @param {Number} sign (int)
* @returns {Number}
* @memberof JXG.Math.ProbFuncs
*/
expx2: function (x, sign) {
// double x;
// int sign;
var u, u1, m, f;
x = Math.abs(x);
if (sign < 0) {
x = -x;
}
// Represent x as an exact multiple of M plus a residual.
// M is a power of 2 chosen so that exp(m * m) does not overflow
// or underflow and so that |x - m| is small.
m = this.MINV * Math.floor(this.M * x + 0.5);
f = x - m;
// x^2 = m^2 + 2mf + f^2
u = m * m;
u1 = 2 * m * f + f * f;
if (sign < 0) {
u = -u;
u1 = -u1;
}
if (u + u1 > this.MAXLOG) {
return Infinity;
}
// u is exact, u1 is small.
u = Math.exp(u) * Math.exp(u1);
return u;
},
/**
*
* Evaluate polynomial
*
* ```
* SYNOPSIS:
*
* int N;
* double x, y, coef[N+1], polevl[];
*
* y = polevl( x, coef, N );
*
* DESCRIPTION:
*
* Evaluates polynomial of degree N:
*
* 2 N
* y = C + C x + C x +...+ C x
* 0 1 2 N
*
* Coefficients are stored in reverse order:
*
* coef[0] = C , ..., coef[N] = C .
* N 0
*
* The function p1evl() assumes that coef[N] = 1.0 and is
* omitted from the array. Its calling arguments are
* otherwise the same as polevl().
*
*
* SPEED:
*
* In the interest of speed, there are no checks for out
* of bounds arithmetic. This routine is used by most of
* the functions in the library. Depending on available
* equipment features, the user may wish to rewrite the
* program in microcode or assembly language.
* ```
*
* @private
* @param {Number} x
* @param {Number} coef
* @param {Number} N
* @returns {Number}
* @memberof JXG.Math.ProbFuncs
*/
polevl: function (x, coef, N) {
var ans, i;
if (Type.exists(coef.reduce)) {
return coef.reduce(function (acc, c) {
return acc * x + c;
}, 0);
}
// Polyfill
for (i = 0, ans = 0; i <= N; i++) {
ans = ans * x + coef[i];
}
return ans;
},
/**
* Evaluate polynomial when coefficient of x is 1.0.
* Otherwise same as polevl.
*
* @private
* @param {Number} x
* @param {Number} coef
* @param {Number} N
* @returns {Number}
* @memberof JXG.Math.ProbFuncs
*/
p1evl: function (x, coef, N) {
var ans, i;
if (Type.exists(coef.reduce)) {
return coef.reduce(function (acc, c) {
return acc * x + c;
}, 1);
}
// Polyfill
for (i = 0, ans = 1; i < N; i++) {
ans = ans * x + coef[i];
}
return ans;
},
/**
*
* Normal distribution function
*
* ```
* SYNOPSIS:
*
* y = ndtr( x );
*
* DESCRIPTION:
*
* Returns the area under the Gaussian probability density
* function, integrated from minus infinity to x:
*
* x
* -
* 1 | | 2
* ndtr(x) = --------- | exp( - t /2 ) dt
* sqrt(2pi) | |
* -
* -inf.
*
* = ( 1 + erf(z) ) / 2
* = erfc(z) / 2
*
* where z = x/sqrt(2). Computation is via the functions
* erf and erfc with care to avoid error amplification in computing exp(-x^2).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -13,0 30000 1.3e-15 2.2e-16
*
*
* ERROR MESSAGES:
*
* message condition value returned
* erfc underflow x > 37.519379347 0.0
* ```
*
* @param {Number} a
* @memberof JXG.Math.ProbFuncs
* @returns {Number}
*/
ndtr: function (a) {
// a: double, return double
var x, y, z;
x = a * this.SQRTH;
z = Math.abs(x);
if (z < 1.0) {
y = 0.5 + 0.5 * this.erf(x);
} else {
y = 0.5 * this.erfce(z);
/* Multiply by exp(-x^2 / 2) */
z = this.expx2(a, -1);
y = y * Math.sqrt(z);
if (x > 0) {
y = 1.0 - y;
}
}
return y;
},
/**
* @private
* @memberof JXG.Math.ProbFuncs
* @param {Number} a
* @returns {Number}
*/
_underflow: function (a) {
if (a < 0) {
return 2.0;
}
return 0.0;
},
/**
*
* Complementary error function
*
* ```
* SYNOPSIS:
*
* double x, y, erfc();
*
* y = erfc( x );
*
*
*
* DESCRIPTION:
*
*
* 1 - erf(x) =
*
* inf.
* -
* 2 | | 2
* erfc(x) = -------- | exp( - t ) dt
* sqrt(pi) | |
* -
* x
*
*
* For small x, erfc(x) = 1 - erf(x); otherwise rational
* approximations are computed.
*
* A special function expx2.c is used to suppress error amplification
* in computing exp(-x^2).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,26.6417 30000 1.3e-15 2.2e-16
*
*
* ERROR MESSAGES:
*
* message condition value returned
* erfc underflow x > 9.231948545 (DEC) 0.0
* ```
*
* @memberof JXG.Math.ProbFuncs
* @param {Number} a
* @returns {Number}
*/
erfc: function (a) {
var p, q, x, y, z;
if (a < 0.0) {
x = -a;
} else {
x = a;
}
if (x < 1.0) {
return 1.0 - this.erf(a);
}
z = -a * a;
if (z < -this.MAXLOG) {
return this._underflow(a);
}
z = this.expx2(a, -1); // Compute z = exp(z).
if (x < 8.0) {
p = this.polevl(x, this.P, 8);
q = this.p1evl(x, this.Q, 8);
} else {
p = this.polevl(x, this.R, 5);
q = this.p1evl(x, this.S, 6);
}
y = (z * p) / q;
if (a < 0) {
y = 2.0 - y;
}
if (y === 0.0) {
return this._underflow(a);
}
return y;
},
/**
* Exponentially scaled erfc function
*
* ```
* exp(x^2) erfc(x)
* valid for x > 1.
* Use with ndtr and expx2.
* ```
*
* @private
* @param {Number} x
* @returns {Number}
* @memberof JXG.Math.ProbFuncs
*/
erfce: function (x) {
var p, q;
if (x < 8.0) {
p = this.polevl(x, this.P, 8);
q = this.p1evl(x, this.Q, 8);
} else {
p = this.polevl(x, this.R, 5);
q = this.p1evl(x, this.S, 6);
}
return p / q;
},
/**
* Error function
*
* ```
* SYNOPSIS:
*
* double x, y, erf();
*
* y = erf( x );
*
*
*
* DESCRIPTION:
*
* The integral is
*
* x
* -
* 2 | | 2
* erf(x) = -------- | exp( - t ) dt.
* sqrt(pi) | |
* -
* 0
*
* For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
* erf(x) = 1 - erfc(x).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,1 14000 4.7e-17 1.5e-17
* IEEE 0,1 30000 3.7e-16 1.0e-16
* ```
*
* @param {Number} x
* @returns {Number}
* @memberof JXG.Math.ProbFuncs
*/
erf: function (x) {
var y, z;
if (Math.abs(x) > 1.0) {
return 1.0 - this.erfc(x);
}
z = x * x;
y = (x * this.polevl(z, this.T, 4)) / this.p1evl(z, this.U, 5);
return y;
},
s2pi: 2.50662827463100050242, // sqrt(2pi)
// approximation for 0 <= |y - 0.5| <= 3/8 */
P0: [
-5.99633501014107895267e1, 9.80010754185999661536e1, -5.66762857469070293439e1,
1.39312609387279679503e1, -1.23916583867381258016
],
Q0: [
1.95448858338141759834, 4.67627912898881538453, 8.63602421390890590575e1,
-2.25462687854119370527e2, 2.00260212380060660359e2, -8.20372256168333339912e1,
1.59056225126211695515e1, -1.18331621121330003142
],
// Approximation for interval z = sqrt(-2 log y ) between 2 and 8
// i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
P1: [
4.05544892305962419923, 3.15251094599893866154e1, 5.71628192246421288162e1,
4.408050738932008347e1, 1.46849561928858024014e1, 2.18663306850790267539,
-1.40256079171354495875e-1, -3.50424626827848203418e-2, -8.57456785154685413611e-4
],
Q1: [
1.57799883256466749731e1, 4.53907635128879210584e1, 4.1317203825467203044e1,
1.50425385692907503408e1, 2.50464946208309415979, -1.42182922854787788574e-1,
-3.80806407691578277194e-2, -9.33259480895457427372e-4
],
// Approximation for interval z = sqrt(-2 log y ) between 8 and 64
// i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
P2: [
3.2377489177694603597, 6.91522889068984211695, 3.93881025292474443415,
1.33303460815807542389, 2.01485389549179081538e-1, 1.23716634817820021358e-2,
3.01581553508235416007e-4, 2.65806974686737550832e-6, 6.2397453918498329373e-9
],
Q2: [
6.02427039364742014255, 3.67983563856160859403, 1.37702099489081330271,
2.1623699359449663589e-1, 1.34204006088543189037e-2, 3.28014464682127739104e-4,
2.89247864745380683936e-6, 6.79019408009981274425e-9
],
/**
*
* Inverse of Normal distribution function
*
* ```
* SYNOPSIS:
*
* double x, y, ndtri();
*
* x = ndtri( y );
*
* DESCRIPTION:
*
* Returns the argument, x, for which the area under the
* Gaussian probability density function (integrated from
* minus infinity to x) is equal to y.
*
*
* For small arguments 0 < y < exp(-2), the program computes
* z = sqrt( -2.0 * log(y) ); then the approximation is
* x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
* There are two rational functions P/Q, one for 0 < y < exp(-32)
* and the other for y up to exp(-2). For larger arguments,
* w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0.125, 1 5500 9.5e-17 2.1e-17
* DEC 6e-39, 0.135 3500 5.7e-17 1.3e-17
* IEEE 0.125, 1 20000 7.2e-16 1.3e-16
* IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ndtri domain x <= 0 -MAXNUM
* ndtri domain x >= 1 MAXNUM
* ```
*
* @param {Number} y0
* @returns {Number}
* @memberof JXG.Math.ProbFuncs
*/
ndtri: function (y0) {
var x, y, z, y2, x0, x1, code;
if (y0 <= 0.0) {
//console.log("ndtri", "DOMAIN ");
return -Infinity; // -this.MAXNUM;
}
if (y0 >= 1.0) {
// console.log("ndtri", 'DOMAIN');
return Infinity; // this.MAXNUM;
}
code = 1;
y = y0;
if (y > 1.0 - 0.13533528323661269189) {
// 0.135... = exp(-2)
y = 1.0 - y;
code = 0;
}
if (y > 0.13533528323661269189) {
y = y - 0.5;
y2 = y * y;
x = y + y * ((y2 * this.polevl(y2, this.P0, 4)) / this.p1evl(y2, this.Q0, 8));
x = x * this.s2pi;
return x;
}
x = Math.sqrt(-2.0 * Math.log(y));
x0 = x - Math.log(x) / x;
z = 1.0 / x;
if (x < 8.0) {
// y > exp(-32) = 1.2664165549e-14
x1 = (z * this.polevl(z, this.P1, 8)) / this.p1evl(z, this.Q1, 8);
} else {
x1 = (z * this.polevl(z, this.P2, 8)) / this.p1evl(z, this.Q2, 8);
}
x = x0 - x1;
if (code !== 0) {
x = -x;
}
return x;
},
/**
* Inverse of error function erf.
*
* @param {Number} x
* @returns {Number}
* @memberof JXG.Math.ProbFuncs
*/
erfi: function (x) {
return this.ndtri((x + 1) * 0.5) * this.SQRTH;
}
};
export default Mat.ProbFuncs;